Hood College
1.0 Definitions And Notations
A. Definitions
(1) Algebra is a systematic investigation of general numbers (algebraic terms) and their relationships. General numbers are letter symbols (A, B, C, ... ; a, b, c,... ; α, β, γ, ...) representing constant or variable quantities.
(2) Algebraic term is a product or quotient of one or more general numbers and of a numerical factor (which can be any natural number) with a prefixed sign (plus sign is frequently omitted). Examples:
__
2a, 3x2, -5x2/y, -√ab, ex are algebraic terms.
(3) Algebraic expression is a collection of one or more algebraic terms connected by the symbols of relationship (See below),
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and/or signs of aggregations (See below)
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and/or signs of operation (See below).
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Examples:
ax2 + bx + c, (a + b)(c + d), ax ÷ b, a > b, b < c, c = d, d ≠ e,
_____
a + bx [(a + b)/c + dx)]n , n/a + bx
c + dx , √ c + dx are algebraic expressions.
(4) Multinomial is an algebraic expression consisting of more than one term. Special cases of multinomials are the binomial (two terms), the trinomial (three terms), and so on. A monomial consists of one term only.
Examples:
Ax2 + bx + c, ax + by + cz, ... are trinomials.
ax + b, ax + by, 3x2y + 2xy, ... are binomials, but they are also multinomials.
(5) Polynomial is a multinomial in which every term is integral and rational in literals. Examples:
5x5 + 3x3y + x3y, 4ab + 3cd + 8ef are polynomials, but
__ __ __
5/x5 + 3y/x4 + x3/y, 4a/b + 3c/d +8e/f, 4√ab + 3√cd + 8√ef are not polynomials.
B. Algebraic Operations
(1) Algebraic transformations involve a finite number of binary operations, governed by algebraic laws [See 2.0 C. below] and rules of signs [See 2.0 D. below).
(2) Four basic algebraic operations are: addition [See 2.0 A. below] and its inverse, subtraction [See 2.0 B. below]; and multiplication [See 2.0 C. below] and its inverse, division [See 2.0 D. below].
(3) Three higher algebraic operations are: involution (See. 10.0 A.) and its two inverses, evolution (See 11.0 A.) and finding the logarithm (See 13.0 A).
C. Algebraic Laws
1) Commutative law
a + b = b + a ab = ba
2) Associative law
a + (b + c) = (a + b) + c a(bc) = (ab)c
3) Distributive law
a(b + c) = ab + ac (a + b)c = ac + bc
4) Division law
If ab = 0, then a = 0 and/or b = 0
D. Rules of Signs
1) Summation
a + (+b) = a + b a + (-b) = a - b
a – (+b) = a – b a – (-b) = a + b
(2) Multiplication
(+a)(+b) = + ab (+a)(-b) = -ab
(-a)(+b) = -ab (-a)(-b) = +ab
(3) Division
(+a) ÷ (+b) = + (a ÷ b) (+a) ÷ (-b) = -(a ÷ b)
(-a) ÷ (+b) = -(a ÷ b) (-a) ÷ (-b) = +(a ÷ b)
2.0 Addition And Subtraction
A. Addition
(1) Addition is the operation of finding the sum of two or several terms. Only like terms can be added. Example:
2a + 4b + 6c + 7a + 8b = 9a +12b + 6c
(2) Order of terms in addition may be changed without affecting the sum (commutative law). Example:
a + b + c = b + c + a = c + a + b, etc.
(3) Grouping of terms in addition may be changed without affecting the sum (associative law). Example:
A + (b + c) = (a + b) + c = (a + c) + b
B. Subtraction
(1) Subtraction is the operation of finding the difference of two terms or of two quantities. Example:
(2a + 4b + 6c) - (7a + 8b) = 2a + 4b + 6c - 7a - 8b = -5a -4b + 6c
(2) Difference of two equal terms is zero. Example:
5x -5x = 0
3.0 Multiplication
A. Basic Products
(1) Multiplication is the operation of finding the product of two or more terms. Example:
2ab * 3bc * 4cd = 24ab2c2d
where * is the algebraic symbol of multiplication, which shall be omitted hereafter.
(2) Order of terms in multiplication may be changed without affecting the product (commutative law). Example:
(2ab)(3bc)(4cd) = (3bc)(4cd)(2ab) = (4cd)(2ab)(3bc)
(3) Grouping of terms in multiplication may be changed without affecting the product (associative law). Example:
[(2ab)(3bc)](4cd) = (2ab)[(3bc)(4cd)]
B. Special Products
(1) Multiplication by a factor.
(a - b + c - d )(+ m ) = am – bm + cm - dm
(a – b + c - d)(-m) = -am +bm -cm +dm
(2) Products of binomials.
(a + b)(c + d) = ac + bc + ad + bd
(a + b)(c - d) = ac +bc - ad – bd
(a - b)(c + d) = ac – bc + ad – bd
(a - b)(c - d) = ac - bc – ad + bd
(a + m)(a + n) = a2 + (m + n)a + mn
(a + m)(a - n) = a2+(m - n)a - mn
(a + b)(a + b) = a2 + 2ab + b2
(a+b)(a-b) = a2 - b2
(a-b)(a-b) = a2-2ab + b2
4.0 Highest Common Factor and Lowest Common Multiple
A. Definitions
(1) Factors of a given algebraic expression are two or more algebraic expressions, the product of which is the given expression.
2) Common factor of two or more expressions is a factor of each of these expressions.
Examples:
ab + ad = a(b + d) a is the common factor
a2b2c + ab2c2 = ab2c(a + c) ab2c is the common factor
3) Prime factor is an algebraic expression divisible by no other expression than itself and 1.
Example:
a (b + c) a and (b + c) are the prime factors
(4) Multiple of a factor is an algebraic expression divisible by this factor.
(5) Common multiple of two or more factors is an expression divisible by each of these factors.
5) Highest common factor (HCF) of two or more expressions is the expression of the highest degree and largest numerical coefficients which is a factor of each of these expressions.
6) Example:
4(a + b)8, 2(a + b)5, 4(a + b)3 2(a + b)3 is the HCF
(7) Lowest common multiple (LCM) of two or more factors is an algebraic expression of the lowest degree and smallest numerical coefficients divisible by each of these factors. Example:
4(a + b)8, 2(a + b)5, 4(a + b)3 4(a + b)" is the LCM
B. Factoring
(1) Factoring into prime factors. If the given algebraic expressions can be factored into prime factors, their HCF and LCM can be determined at once from these factors.
(2) Highest common factor is the product obtained by taking each factor to the lowest power to which it occurs in any of these expressions.
(3) Lowest common multiple is the product obtained by taking each factor to the highest power to which it occurs in any of these expressions.
(4) The process of factoring an algebraic expression is also known as decomposition.
5.0 Decomposition Of Binomials And Trinomials
A. General Cases
(1) Decomposition of binomials, zero remainder (k = 1, 2, 3, ..).
a2k – b2k = (a + b)( a2k-1 ± a2k-1b +2a2k-3 ± ... ±b2k-1)
a2k+1 – b2k+1 = (a - b)(a2k + a2k-1b + a2k-2b2 + … b2k)
a2k+1 + b2k+1 = (a + b)(a2k - a2k-1b + a2k-2b2 - … b2k)
(2) Decomposition of binomials, with remainder (k = 1, 2, 3, ...).
a2k + b2k = (a + b)(a2k-1 ± a2k--2b + a2k-3b2" + … b2k-1 + 2b2k
a2k+1 – b2k+1 = (a + b)(a2k – a2k-1b2 + a2k-2b2 - … + bk) – 2b2k+1
a2k+1 + b2k+1 = (a -b)(a2k +a2k-1b2 + a2k-2b2 +… + bk) +2b2k+1
(3) Decomposition of binomials into products of binomials.
a2k – b2k = (ak + bk)(ak – bk)
a2k + b2k = (ak +bk)(ak - bk) + 2b2k
(4) Decomposition of trinomials.
x2 + px + q = (x + m)(x + n)
where p = m + n and q = mn.
Example:
x2 - 5x - 14 = (x - 7)(x + 2)
B. Special Cases
(1) Decomposition of even-power binomials (a2k – b2k).
a2 – b2 = (a - b)(a + b)
a4 – b4 = (a-b)(a3 + a2b + ab2 + b3)
a6 – b6 = (a - b)(a5 + a4b + a3b2 + a2b3 + ab3 + b5)
a4 – b4 = (a + b)(a3 – a2b + ab2 – b2)
a6 – b6 = (a + b)(a5 – a4b + a3b2 - a2b3 + ab4 – b5)
(2) Decomposition of even-power binomials (a2k + b2k).
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a2 + b2 = (a + √2ab + b)(a - √2ab +b)
_ _
a4 + b4 = (a2 + ab√2 + b2)(a2 - ab√2 + b)
____ ____
a6 + b6 = (a3 + √2a3b3 + b3)(a3 - √2a3b3 + b3)
(3) Decomposition of odd-power binomials (a2k+1 – b2k+1)
a3 – b3 = (a – b)(a2 + ab + b2)
a5 – b5 = (a-b)(a4+ a3b + a2b2 + ab3 + b4)
7) Decomposition of odd-power binomials (a2k+1 +b2k+1)
a5 + b5 = (a+b)(a2 – ab + b2)
a5 + b5 = (a+b)(a4 – a3b + a2b2 – ab3 + b4)
6.0 Division
A. Basic Quotients
(1) Division is the operation of finding the quotient of two terns. Examples:
6a2b ÷ 3ab = 2a
8(a2+ ab) ÷ 4a = 2(a + b)
(2) Quotient of two equal terms is 1. Example:
3abc ÷ 3abc = 1
B. General Quotients
(1) Binomials, zero remainder (k = 1, 2, 3, … ,).
(a2k – b2k) ÷ (a ± b) = a2k-1 ± a2k-2b + a2k-3b3 ± ... ± b2k-1
(a2k+1 – b2k+1) ÷ (a – b) = a2k + a2k-1b + a2k-2b2 + … + b2k
(a2k+1 + b2k+1) ÷ (a + b) = a2k - a2k-1b + a2k-2b2 + … + b2k
(a2k – b2k) ÷ (ak + bk) = ak - bk
(a2k – b2k) ÷ (ak - bk) = ak + bk
(2) Binomials, with remainder (k = 1, 2, 3, …).
(a2k – b2k) ÷ (a ± b) = a2k-1 ± a2k-2b2 ± … b2k-1 + 2b2k/(a ± b)
(a2k+1 – b2k+1) ÷ (a + b) = a2k - a2k-1b + a2k-2b2 - … + b2k – 2b2k+1/(a + b)
(a2k+1 + b2k+1) ÷ (a - b) = a2k + a2k-1b + a2k-2b2 + … + b2k + 2b2k+1/(a + b)
C. Special Quotients
(1) Even-power binomials.
(a2 – b2) ÷ (a + b) = a – b
(a2 – b2) ÷ (a - b) = a + b
(a4 – b4) ÷ (a + b) = a3 – a2b + ab2 – b3
(a4 – b4) ÷ (a - b) = a3 + a2b + ab2 + b3
(a6 - b6) ÷ (a + b) = a5 – a4b + a3b2 – a2b3 + ab4 – b5
(a6 - b6) ÷ (a - b) = a5 + a4b + a3b2 + a2b3 + ab4 + b5
(a4 – b4) ÷ (a2 + b2) = a2 – b2
(a4 - b4) ÷ (a2 – b2) = a2 + b2
(a6 – b6) ÷ (a3 + b3) = a3 – b3
(a6 - b6) ÷ (a3 – b3) = a3 + b3
(2) Odd-power binomials.
(a3 – b3) ÷ (a - b) = a2 + ab + b2
(a3 + b3) ÷ (a + b) = a2 – ab + b2
(a5 – b5) ÷ (a - b) = a4 +a3b + a2b2 + ab3 + b4
(a5 +b5) ÷ (a +b) = a4 -3b + a2b2 -ab3 + b4
7.0 Simple Fractions
A. Definitions
(1) Algebraic fraction is an indicated division of two algebraic expressions called again the numerator N and denominator D. Examples:
2ab , a2 – b2 , ___3+x___ are rational algebraic fractions.
3b a + b x2 + 6x + 8
(2) Simple and complex fractions. When N and D are integral expressions, the fractions are termed simple fractions. When N or D or both are simple fractions, the fraction is termed a complex fraction. Examples:
1 , a , a2 + b2 are simple fractions. 1 , a/b (a2 + b2)/c are complex fractions.
a b c + d a/b c (d + e)/f2
B. Principles Used in Operations
(1) Enlarging of fraction by multiplying both its numerator and its denominator by the same expression (except zero) does not change the value of the fraction.
Examples:
a = am x + 2 = (x + 2)(x + 4) = x2 + 6x + 8
b bm x + 3 (x + 3)(x + 4) x2 + 7x + 12
(2) Reduction of fraction by dividing both its numerator and its denominator by the same expression (except zero) does not change the value of the fraction.
Examples:
am = a x2 + 6x + 8 = (x + 2)(x + 4) = x + 2
bm b x2 + 7x + 12 (x + 3)(x + 4) x + 3
where the common factors in the term of the numerator and the identical term in the denominator are removed.
(3) Change in sign of both numerator and denominator of a fraction does not change the sign of the fraction.
Examples:
a/b = -a/-b -a/b = a/-b = -(a/b) -1/(c – d) = a/(d-c) = -[a/(c-d)]
(4) Change In sign of either numerator or denominator changes the sign of the fraction.
Examples:
a/b ≠ -a/b a/b ≠ a/-b a/b ≠ -(a/b) a –b/c – d ≠ -[(a – b)/(c – d)
(5) Lowest common denominator (LCD) of two or more fractions is the lowest common multiple of their denominators.
Example:
LCD of 1/a, 1/(x + 2), 1/(xx + 5x +6) is a(x + 2)(x + 3)
Their transformation to this denominator accomplished by rule (1) of this section is, since x2 + 5x +6 = (x +2)(x +3),
1/a = (x +2)(x +3)/a(x +2)(x +3)
1/(x + 2) = a(x + 3)/a(x +2)(x +3)
1/x2 + 5x +6 = a/a(x +2)(x +3)
C. Addition and Subtraction
(1) Algebraic fractions of common denominators can be added or subtracted.
4) Sum or difference of two algebraic fractions with a common denominator equals a fraction whose numerator is the sum or difference of their numerators and whose denominator is their common denominator.
5)
Example:
a/m ± b/m = (a ± b)/m
(3) Sum and difference of two algebraic fractions of different denominators is found by transforming the given fractions to their LCD.
Examples:
a/m ± b/n = (am ± bm)/mn a/m ± b/mn = (an ±b)/mn
a/m ± b/n ± c/p (anp ± nmp ± cmn)/mnp
a/(1 + x) + b/(1 – x) + c/(1 – x2) = (a – ax + b + bx + c)/ (1 –x2) =
a + b + c – x(a – b)/(1-x2)
(4) Sum and difference of an algebraic expression and of an algebraic fraction is found by transforming the algebraic expression to a fraction of the same denominator.
Example:
A ±b/m = am/m ± b/m = (am ±b)/m
D. Multiplication and Division
(1) Product of two or more algebraic fractions is the product of their numerators divided by the product of their denominators.
Examples:
a/m x b/n = ab/mn (x + 1)/(x - 1) x (x + 1)/(x - 2) = (x + 1)(x + 1)/(x2 + 2x +1)
(x + 1) (x - 2) (x + 3) = (x + 3)
(x - 1) (x + 1) (x – 2) (x – 1)
where (x +1) and (x + 2) are common factors in the term of the numerator and the identical term in the denominator are removed.
(2) Product of an algebraic expression and an algebraic fraction is a fraction whose numerator is the product of the expression and the numerator of the fraction and whose denominator is the denominator of the fraction.
Examples:
a (b/m) = (ab)/m a(x + b)/m = (ax + ab)/m
(3) Reciprocal of an algebraic expression is 1 divided by this expression. Reciprocal of an algebraic fraction is a fraction obtained by reversing the positions of the numerator and denominator in the given fraction.
Examples:
The reciprocal of a is 1/a . Reciprocal of a/b is b/a.
(4) Product of an algebraic expression and its reciprocal is 1. Product of an algebraic fraction and its reciprocal is also 1.
Examples:
a(1/a) = 1 (a/b) x (b/a) = 1
(5) Quotient of two or more algebraic fractions is the product of the first fraction and the reciprocals of the remaining fractions.
Examples:
(a/m) ÷ (b/n) = (a/m) x (n/b) = (an/bm) [(a/m) ÷ (b/n)] ÷ (c/p) =
(a/m) x (n/b) x (p/c) = (anp/bcm)
where the brackets indicate that the whole quantity (a/m) ÷ (b/n) is to be divided by c/p.
(6) Quotient of an algebraic fraction and an algebraic expression is the fraction whose numerator was divided by the expression or whose denominator was multiplied by the same expression.
Examples:
(a/m) ÷ b = (a ÷ b)/m (a/m) ÷ b = a/(bm)
(a2 – b2)/m ÷ (a – b) = (a2 – b2)/m(a – b) = (a + b)(a – b)/m(a – b) = (a + b)/m
where (a - b) has been canceled out.
8.0 Complex, Compound, And Continued Fractions
A. Complex Fractions
(1) Definition of complex algebraic fraction is identical to that of complex fraction in arithmetic.
(2) Reduction procedure of a complex algebraic fraction as illustrated symbolically below.
Examples:
(a/b)/(c/d) = (ad) / (cd) [(am)/(bn)]/[(cm)/(dn)] = (admn)/(bcmn) = ad/bc
(a/b)/c = (a/b)/(c/1) = a / (bc) a / (c/d) = (a/1)/(c/d) = ad/c
(3) Operations. Once the complex algebraic fraction is reduced to a simple algebraic fraction. The same applies in cases of compound and continued fractions.
Examples:
(a/m)/(b/n) ± (c/p)/(d/q) = (an/bm) x (cq/dp) = (acnq)/(bdmp)
(a/m)/(b/n) x (c/p)/(d/q) = (an/bm) x (cq/dp) = (acnq)/(bdmp)
(a/m)/(b/n) ÷ (c/p)/(d/q) = (an/bm) ÷ (cq)/(dp) = (adnp)/(bcmq)
B. Compound Fractions
(1) Definition of a compound algebraic fraction is identical to that of arithmetic compound fraction.
(2) Reduction procedure of compound algebraic fraction is as illustrated symbolically below.
Examples:
[(a ± b/m)]/[c ± (d/n)] = [(am ± b)/m]/[(cn ± d)/n] = [n(am ± b)]/[m(cn ± d)]
(a ± (b/m)/c = [(am ± b)/m]/(c/1) = [(am ± b)]/cm
a/[c ± (d/n)] = (a/1)/[(an ± d)]/n = an/(cn ±d)
C. Continued Fractions
(1) Definition of continued algebraic fraction is identical to that of arithmetic compound fraction.
(2) Reduction procedure of continued algebraic fraction is as illustrated symbolically below.
Example:
a + _____1_____ = a + ______1_____ = a + ____1____
b + ___1___ b + ___1/1___ b + ___d___
c + 1/d (cd + 1)/d (1 + cd)
1
= a + 1_ ___ = a + ___1 + cd___
b(1 + cd) + d d + b(1 + cd)
1 + cd
= ad + ab(1 + cd) + 1 + cd = ad + (1 + ab)(1 + cd)
d + b(1 + cd) d + b(1 + cd)
9.0 Operations With Zero
A. Addition and Subtraction
1) Sum of an algebraic expression and zero equals the expression.
Examples:
a + 0 = a 0 + a = a
(2) Difference of an algebraic expression and zero equals the expression.
Examples:
a – 0 = a but 0 - a = -a
B. Multiplication and Division
1) Product of an algebraic expression and zero equals zero.
Examples:
a x 0 = 0 0 x a = 0
(2) Quotient of an algebraic expression and zero is undefined but a quotient of zero and of an algebraic expression is zero.
Examples:
a ÷ 0 = indeterminate 0 ÷ a = 0
10.0 Exponents
A. Definitions
(1) Positive Integral exponent. If n is a positive integer and a is an algebraic expression, then
an = a x a x a x a … a
n times
is said to be the nth power of a, n is the exponent of the power, and a is the base.
(2) Negative integral exponent. If -n is a negative integer and a is an algebraic expression, then
a-n = (1/a) x (1/a) x (1/a) x (1/a) x … (1/a) = (1/an)
n times
is said to be the nth power of the reciprocal of a.
(3) Rules of signs. If 2n is an even integer and 2n + 1 is an odd integer, then
(± a)2n = a2n (± a)2n+1 = ± a2n+1
(4) Special cases. if a ≠ 0, then
a0 = 1 a' = a (1/a0) = 1 (1/a1) = (1/a)
B. Laws of Exponents (m, n = Integers)
(1) Addition and subtraction (p, q = algebraic expressions).
pam ± qam = (p ± q)am
pa-m ± qa-m = (p ± q)a-m = (p ± q)/am
2) Multiplication and division.
aman = am+n am ÷ an = am-n ama-n = am-n am ÷ a-n = am+n
a-m ÷ an = am-n a-m x a-n = amn a-m ÷ a-n = a-m+n
3) Powers of products and quotients.
(ab)m = ambm (a ÷ b)m = am ÷ bm
(ab)-m = 1/(ambm) (a ÷ b)-m = bm ÷ a m
(4) Powers of powers and fractions.
(am)n = (an)m = amn (a/b)m = am/bm
(am)-n = (a-m)n = 1/amn (a/b)-m = bm/am
C. Powers of Binomials and Trinomials
(1) Binomials.
(a ± b)2 = a2 ± 2ab + b2
(a ± b)3 = a3 ± 3a2b + 3ab2 ± b3
(a ± b)4 = a4 ± 4a3b + 6a2b2 ± 4ab3 + b4
(a ± b)5 = a5 ± 5a4b + l0a3b2 ± l0a2b3 + 5ab4 + b5
(2) Trinomials.
(a ± b + c)2 = a2 + b2 +c 2 ± 2ab ± 2bc + 2ac
(a ± b + c)3 = a3 ± b3 + c3 + 3a2(±b + c) +3b(a + c) + 3c2(a ± b) ± 6abc
11.0 Radicals
A. Definitions
(1) Positive Integral Index. If n is a positive integer and a is an algebraic expression, then
__
r = n√ a = a1/n
is said to be the radical (nth root) of a, which must satisfy the relationship
a = r x r x r x r … r = rn
n times
where a is the radicand (base) and n is the index of the radical.
_
2) Square root. When n = 2, the radical of a is called a square root of a, r = √a and the
_ _
index 2 is omitted; that is, 2√a - √a.
(3) Negative Integral index. If - n is a negative integer and a is again an algebraic expression, then
_ _
1/n = -n√a = 1/ n√a = 1/(a1/n)
is said to be the reciprocal of the radical (nth root) of a, which must satisfy the relationship
1/a = 1/r x 1/r x 1/r x 1/r … 1/r = 1/rn
n times
where a is again the radicand (base) and -n is the negative index of the radical.
(4) Rational number. When a in r is the nth power of a real number, then r is a rational number.
(5) Irrational number. When a in r is not the nth power of a real number, then r is an irrational number.
14.0 Linear Equations In One Unknown
A. Definitions
(1) Equation is a statement of equality of two expressions.
(2) Equations are classified as identities, conditional equations, and functional equations.
3) Identity is the equality of two constant terns. examples: A = B 1+2=3
4) Conditional equations contain one or several unknowns.
Examples:
ax + b = 0 ax + by +cz + d = 0
where x, y, z are the unknowns.
5) Functional equations (functions) state the relationship between two or several variables.
Examples:
y = ax + b y = a0 + a1x1 + a2x0 + . . .
where y changes as x changes or as x1 x2 ... change.
(6) Algebraic equation of nth degree In the unknown x is a0 + a1x +a2x2 + … + an-1xn-1 + anxn
in which a0, a1, a2, ... are real or complex quantities.
(7) Linear algebraic equation In one unknown is
ax + b = 0 where a ≠ 0.
B. Solution
(1) Roots of equation. To solve an algebraic equation for the unknown x means to find values of x (roots of equation) which satisfy this equation.
(2) System of n algebraic equations. A set of n equations which are valid only for a certain definite set of values of the unknowns x1, x2, ..., xn is called a system of n equations. Such a set of values is called the solution of this system.
C. Axioms of Solution
(1) Equivalent equations. Two equations that have the same roots are said to be equivalent equations.
(2) Addition or subtraction of equal terms. If the same term is added to or subtracted from both sides of a given equation, the new equation is equivalent to the given equation (has the same roots).
Example, addition:
Given equation x - a = b
Addition of a + a = + a
Equivalent equation x = b + a
This operation is equivalent to the carrying of term a with the opposite sign to the other side of the equation.
Example, subtraction:
Given equation x + c = d
Subtraction of c - c = -c
Equivalent equation x = d -c
This operation is also equivalent to the carrying of term c with the opposite sign to the other side of the equation.
(3) Multiplication or division by equal terms. If both sides of an equation are multiplied or divided by the same term (not zero), the new equation is equivalent to the original equation (has the same roots).
Example, multiplication:
Given equation x = b + c
b d
Multiplication by a x a = b + c a
b d
Equivalent equation x = a(b + c)
d
This procedure is also equivalent to the carrying of term a from the denominator of the other side.
Example, division:
Given equation qx = g + h
k
Division by q qx = g + h
q qk
Equivalent equation x = g + h
qk
This procedure is also equivalent to the carrying of term q from the numerator of one side to the denominator of the other side.
15.0 Linear Equations In Two Unknowns
A. Definitions
(1) Independent equations. Two linear equations in two unknowns x and y.
A1x + b1y = c1
a2x + b2y = c2
are independent if neither can be derived from the other by algebraic operations.
(2) Consistent and simultaneous equations. The two equations (above) are said to be consistent and simultaneous if a, b, c are constants, a, b are not both zero in one equation, both c's are not zero, and they are satisfied by one pair of values (one value for x; another value for y).
B. Methods of Solution
(1) Three methods of solution are commonly used for the solution of linear equations in two unknowns: substitution, comparison, and elimination.
(2) Substitution method. First express one of the unknowns from one of the equations. Then substitute this value in the other equation and solve for the second unknown. Finally return to the equivalent first equation and solve for the final value of the first unknown.
Example:
Given a1x + b1y = c2 a2x + b2y = c2
Isolate x x = c1 – b1y
a1
Substitution of x a2 c1 – b1y + b2y = c2
a1
Isolate y y = a1c2 – a2c1
a1b2 – a2b1
x becomes x = c1b2 – c2b1
a1b2 – a2b1
(3) Comparison method. Express the first unknown from each equation and equate their right sides. Then solve this equation for the second unknown. Finally return to the equivalent equation (to the simplest one) and solve for the first unknown.
Example:
Given a1x + b1y = c1 x2 + b2y = c2
Isolating x yields x = c1 – b1y x = c2 – b2y
a1 a2
Equality of the right sides of (above) gives: c1 – b1y = c2 – b2y
a1 a2
Isolating y yields: y = a1c2 – a2c1
a1b2 – a2b1
And x becomes x = c1b2 – c2b1
a1b2 – a2b1
(4) Elimination method. Multiply or divide each equation by such factors as to make equal the coefficients of one unknown in each equation. Then add or subtract these modified equations so as to eliminate one unknown. Solve the resulting equation for the remaining unknown/ Then substitute this value in the simplest one of the original equations and solve the first unknown.
Example:
Given a1x + b1y = c1 a2x + b2y = c2
Divide the a constants in each equation above: x + b1y = c1 x + b2y = c2
a1 a1 a2 a2
Isolate x x + b1y = c1 x = c1 - b1y x + b2y = c2 x = c2 – b2y
a1 a1 a1 a1 a2 a2 a2 a2
When the right sides of the equation are combined: c1 - b1y = c2 – b2y
a1 a1 a2 a2
Reorganizing above: c1 – c2 = b1y – b2y y b1 – b2 = c1 – c2
a1 a2 a1 a2 a1 a2 a1 a2
y = a1c2 – a2c1
a1b2 a2b1
Isolate y and solve for x x = c1b2 – c2b1
a1b2 a2b1
16.0 Proportions
A. Definition of Proportions
(1) Simple proportion is the equality of two ratios.
a:b = c:d a/b = c/d
(2) Successive proportion is the equality of several ratios.
a:b = c:d = e:f = g:h a/b = c/d = e/f = g/h
(3) Arithmetic proportion is defined as
(a- x):(x-b) = c : c (a- x)/(x-b) = c/c
where x is the arithmetic mean of a and b, and c is an arbitrary term (also 1 but not zero).
17.0 Quadratic Equations In One Unknown
A. Standard Case
(1) Quadratic equation in one unknown is an integral rational equation in which the term of highest degree in the unknown is of the second degree.
ax2 +bx + c = 0
where a, b, c are real, a ≠ 0, and x is the unknown.
(2) Roots. The solution of this equation is
_______
x1, 2 = –b ± √ b2 –4ac
2a
where b2 - 4ac, is called the discriminant.
(3) Classification of roots. If
b2 - 4ac > 0, the roots are real and unequal,
b2 - 4ac = 0, the roots are real and equal,
b2 - 4ac < 0, the roots are conjugate complex.
Example: real and unequal roots
Given 2x2 - 4x - 16 = 0
_____________
By formula x1, 2 = –(-4) ± √ (-4)2 –4(2)(-16) = 4 ± 12 = 1 ± 3 x1 = 4, x2 = –2
2(2) 4
Example: real and equal roots
Given 2x2 - 8x - 8 = 0
___________
By formula x1, 2 = –(8) ± √ (8)2 –4(2)(8) = –8 = –2 x1 = x2 = –2
(2)(2) 4
Example: conjugate complex roots
Given 2x2 -3x + 4 = 0
___________ __
By formula x1, 2 = –(-3) ± √(-3)2 –4(2)(4) = 3 ± i√23
(2)(2) 4
__ __
And x1 = 3 + i√23 x1 = 3 - i√23
4 4
__
Please note that "i" is an imaginary number that represents the √-1 , you will have to wait exposure to the concept of imaginary numbers for an explanation of 'i"s use.
B. Special Cases
(1) If b = 0, the reduced quadratic equation is called pure quadratic equation
_____
ax2 + c = 0 and x1, 2 = ± √-(c/a)
where if c/a > 0, the roots are imaginary, equal but of opposite sign, and if c/a < 0, the roots are real, equal but of opposite sign.
(2) If c = 0, the standard form reduces to
ax2 + bx = 0 or x(ax + b) = 0
and x1 = 0 x2 = -(b/a)
C. Biquadratic Equation
(1) Biquadratic equation in one unknown is an integral rational equation in which the unknowns are of the fourth and second degree.
ax4 + bx2 + c = 0
(2) Transformation. By the substitution y = x this equation reduces to the quadratic equation
ay2 + by + c = 0
(3) Roots. The solution of this transformed equation is
_____________
x1, 2, 3, 4 = / _______
√ –b ± √ b2 –4ac
2a
22.0 Pascal's Triangle
A. Coefficients of the binomial expansion can also be represented by a triangle of integers, where the lower number equals the sum of the two adjacent numbers above.
B. Triangle
[pic]
C. Applications
(a ± b)° = 1
(a ± b)1 = a ± b
(a ± b)2 = a2 ± 2ab + b2
(a ± b)3 = a3 ± 3a2b + 3ab2 ± b3
(a ± b)4 = a4 ± 4a3b + 6a2b2 + 4ab3 + b4
(a ± b)5 = a4 ± 5a4b + l0a3b2 + 10a2b3 + 5ab4 ± b5
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